Scott continuity

In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. $\sqcup f[D]=f(\sqcup D)$ , where $\sqcup$ is the directed join.^[1] When $Q$ is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.^[2]

A subset O of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in O have non-empty intersection with O. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.^[1]

The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.^[3]

Scott-continuous functions are used in the study of models for lambda calculi^[3] and the denotational semantics of computer programs.

Properties

A Scott-continuous function is always monotonic, meaning that if $A\leq _{P}B$ for $A,B\subset P$ , then $f(A)\leq _{Q}f(B)$ .

A subset of a directed complete partial order is closed with respect to the Scott topology induced by the partial order if and only if it is a lower set and closed under suprema of directed subsets.^[4]

A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T₀ separation axiom).^[4] However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial.^[4] The Scott-open sets form a complete lattice when ordered by inclusion.^[5]

For any Kolmogorov space, the topology induces an order relation on that space, the specialization order: x ≤ y if and only if every open neighbourhood of x is also an open neighbourhood of y. The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be sober: the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.^[4]

Examples

The open sets in a given topological space when ordered by inclusion form a lattice on which the Scott topology can be defined. A subset X of a topological space T is compact with respect to the topology on T (in the sense that every open cover of X contains a finite subcover of X) if and only if the set of open neighbourhoods of X is open with respect to the Scott topology.^[5]

For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.^[6]

Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.^[7]

Footnotes

^ ^a ^b Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.
^ Scott topology at the nLab
^ ^a ^b Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.
^ ^a ^b ^c ^d Abramsky, S.; Jung, A. (1994). "Domain theory" (PDF). In Abramsky, S.; Gabbay, D.M.; Maibaum, T.S.E. (eds.). Handbook of Logic in Computer Science. Vol. III. Oxford University Press. ISBN 978-0-19-853762-5.
^ ^a ^b Bauer, Andrej & Taylor, Paul (2009). "The Dedekind Reals in Abstract Stone Duality". Mathematical Structures in Computer Science. 19 (4): 757–838. CiteSeerX 10.1.1.424.6069. doi:10.1017/S0960129509007695. S2CID 6774320. Retrieved October 8, 2010.
^ Barendregt, H.P. (1984). The Lambda Calculus. North-Holland. ISBN 978-0-444-87508-2. (See theorems 1.2.13, 1.2.14)
^ N. Belnap (1975) "How Computers Should Think", pages 30 to 56 in Contemporary Aspects of Philosophy, Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6

References

"Scott Topology". PlanetMath.

[Vickers1989-1] Vickers, Steven (1989). Topology via Logic. Cambridge University Press. ISBN 978-0-521-36062-3.

[2] Scott topology at the nLab

[Scott1972-3] Scott, Dana (1972). "Continuous lattices". In Lawvere, Bill (ed.). Toposes, Algebraic Geometry and Logic. Lecture Notes in Mathematics. Vol. 274. Springer-Verlag.

[AbramskyJung1994-4] Abramsky, S.; Jung, A. (1994). "Domain theory" (PDF). In Abramsky, S.; Gabbay, D.M.; Maibaum, T.S.E. (eds.). Handbook of Logic in Computer Science. Vol. III. Oxford University Press. ISBN 978-0-19-853762-5.

[BauerTaylor2009-5] Bauer, Andrej & Taylor, Paul (2009). "The Dedekind Reals in Abstract Stone Duality". Mathematical Structures in Computer Science. 19 (4): 757–838. CiteSeerX 10.1.1.424.6069. doi:10.1017/S0960129509007695. S2CID 6774320. Retrieved October 8, 2010.

[6] Barendregt, H.P. (1984). The Lambda Calculus. North-Holland. ISBN 978-0-444-87508-2. (See theorems 1.2.13, 1.2.14)

[7] N. Belnap (1975) "How Computers Should Think", pages 30 to 56 in Contemporary Aspects of Philosophy, Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6

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